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Unformatted text preview: Problem Set 4: Cournot Model, Product Differentiation, Vertical Integration Question 1 : Consider a Cournot duopoly that sells a homogeneous product. Market demand given by: P ( Q ) = 100- Q , where Q denotes aggregate output. There are two firms, each with cost functions: c ( q ) = 10 q . (i) Define the game being played. Players: each firm. Actions: set output quantity. Payoffs: 1 ( q 1 ,q 2 ) = q 1 (100- q 1- q 2 )- 10 q 1 , 2 ( q 1 ,q 2 ) = q 2 (100- q 1- q 2 )- 10 q 2 (ii) Determine the best response functions. The FOCs will determine the best response functions. 1 ( q 1 ,q 2 ) q 1 = 0 100- 2 q 1- q 2- 10 = 0 q BR 1 ( q 2 ) = 90- q 2 2 . 2 ( q 1 ,q 2 ) q 2 = 0 100- 2 q 2- q 1- 10 = 0 q BR 2 ( q 1 ) = 90- q 1 2 . (iii) Determine firm outputs, aggregate output, and market price under the Cournot-Nash equilibrium. A Nash Equilibrium is an action profile to which all players are best responding. This is equivalent here to looking for the intersection point of the best respond functions. Solve the system: ( q BR 2 ( q 1 ) = 90- q BR 1 2 q BR 1 ( q 2 ) = 90- q BR 2 2 This yields q * 1 = q * 2 = 30 ,Q * = q * 1 + q * 2 = 60 ,P * = P ( Q * ) = 40. (iv) Calculate the deadweight loss. How does it compare to the deadweight loss of a monopolized market? The competitive price-quanitty is Q C = 90 , P C = 10 . Here, the oligopoly price-quantity is Q * = 60 , P * = 40 . Deadweight loss is the forgone social surplus: 1 2 (90- 60) (40- 10) = 450 . The deadweight loss of the monopolized market would be: 1 2 (90- 45) (55- 10) = 1012 . 5 . The duopoly creates less deadweight loss than the monopoly. (v) Who is better off with Cournot competition? Consumers? Producers? Society? Since prices are below the monopoly price, the producers are definitely worse off. Since prices are lower and output is higher, consumers are definitely better off. Since deadweight loss is reduced, society is better off. (vi) Suppose the Cournot competitors could sit down at a table and discuss a price-setting agreement. What is the best price they should agree on? Where is the difficulty in maintaining this agreement? The monopoly price yields the greatest producer surplus possible. If the competitors could agree to choose output levels which sum to the monopoly output level and split the producer surplus, they could do better as a group. For example, if they each agreed to produce one-half of the monopoly output level, they could both be better off. The difficulty lies in the fact that such an action profile is not a Nash Equilibrium (look at figure 8.4) since such action profile do not fall on the best response curves. Firms are not maximizing their own profits under the collusive plan and have short-run incentives to deviate. This problem is like the Prisoners Dilemma....
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- Spring '11